Transformation of thermal parameters

[rkaminski]

Hi All,

Have been looking at the literature but did not found a simple way to do it. How to transform themal parameters expressed as 's or 's which were rotated along some arbitrary axis and arbitrary angle? I mean I have a, for example, rotation matrix along some Cartesian X-axis (I know the transformation from crystallographic system of course). How one should transform thermal parameters respectively?

Thanks in advance,

Radek

[oleg]

Assuming your U is in the Cartesian frame and M is the rotation matrix:

U' = M*U*Mt, where Mt is the transpose of M,

Cheers,

o

[pascalp]

It will work in any basis for a symmetry operation (the coordinate system does not change). So, you don't need to be in a cartesian coordinate system.

For a more general case (change of basis), you need the dimensionless mean-square displacement tensor and the metric tensor.

[rkaminski]

Imagine I would like to change the basis as my rotation matrix is given in the Cartesian system. For example like:



How should I then transform matrix. Is it something like this below?



where transforms coordinates from fractional to Cartesian system.

[pascalp]

is in a cartesian basis
is in the crystal basis
is dimensionless, this is important, as in:

is the inverse
is the transpose
is the inverse transpose

To transform a vector of coordinates using M


To transform a tensor:


Then you apply your rotation:


If you want go back into the crystal system:


After simplification:



I thought I explained this in a publication but not this case, not when the transformation is defined in a diferent basis.
http://scripts.iucr.org/cgi-bin/paper?S0108767311018216

I forgot to mention that this is valid for beta but not for U as in the cif file. There are couple of steps more.

[pascalp]

one more thing, you may have noticed:

the transformed beta tensor is


and the new rotation matrix:


Formula are differents because beta is a doubly contravariant second rank tensor and R a mixed second rank tensor.

[rkaminski]

To the answer below, problem solved:) It was a stupid error in the Excel worksheet. Above transformations work perfectly.

Radek

*********

Thank you. This is basically the result I have obtained a bit later. However, after putting everything in Excel I get ADPs that after visualization the ellipsoids differ a bit in their shape. I always thought that such transformation should retain the principal axes magnitudes. How sensitive are those operations for numerical errors? Or maybe being much too tired I miss some stupid error...

[pascalp]

It is the case, the eigenvalues and eigen vectors remains the same. But the formula is different in the crystal basis. You need to use the general formula:
http://en.wikipedia.org/wiki/Generalize ... ue_problem
A is your tensor, B is the metric tensor.

In python you need to use scipy.linalg.eig and not numpy.linalg.eig I don't think excel can do that.

If the matrices are ill conditioned you can have some problems but it should not be the case.

I can have a look at your xls sheet if your want.